Mathematical formulation of quantum mechanics

The mathematical formulations of quantum mechanics are ways to describe quantum mechanics using math. These ways use parts of functional analysis, like special areas called Hilbert spaces. These are types of linear space.

In quantum mechanics, things we can measure, like energy and momentum, are not just numbers. They are linked to special math ideas called eigenvalues of things called operators in Hilbert space. These ideas about the quantum state are different from older science ideas.

These math ways are still used today. One big idea is that we cannot measure some things at the exact same time. This was first shown by Heisenberg with a thought experiment. Before quantum mechanics, science used older math tools like calculus. Even theories of relativity used these older tools.

History of the formalism

The "old quantum theory" and the need for new mathematics

Main article: Old quantum theory

In the 1890s, Planck found that energy could only be given or taken in small, fixed amounts called quanta. This helped explain some light patterns that the old science could not. In 1905, Einstein said these energy packets, later called photons, were like tiny bits of light.

These ideas made old science look different. Scientists like Bohr and Sommerfeld tried to change classical mechanics to fit the new ideas. But they had trouble with bigger atoms. In 1923, de Broglie said that waves and particles were two ways to look at the same thing, and this worked not just for light but for all matter.

The "new quantum theory"

In the mid-1920s, scientists made new math rules to explain these ideas. Heisenberg made matrix mechanics, and Schrödinger made wave mechanics. Both ways worked well and were later proved to mean the same thing. They used hard math, like Hilbert spaces, to explain how very small parts of the world behave.

These new ideas helped solve many secrets about atoms and started what we now call quantum mechanics.

Postulates of quantum mechanics

Quantum mechanics uses special math to describe how tiny parts of the universe, like atoms and particles, behave. These ideas help scientists make predictions about things we can’t see with our eyes.

The main ideas in quantum mechanics are about three things: the state of a system, how we measure it, and how it changes over time. A system’s state is like a description of everything about it at one moment. In quantum mechanics, this is shown using something called a Hilbert space, which is a special kind of math space. When we measure something, like the spin of an electron, the result depends on the state of the system. And over time, the state changes following rules that scientists can write as equations.

Mathematical structure of quantum mechanics

Main article: Dynamical pictures

Quantum mechanics uses special math to describe how very small parts of the world, like atoms, behave. Instead of using simple numbers, it uses something called Hilbert spaces. These are like fancy math rooms where we can track all the possible states of a particle. This math helps scientists predict how things like light or electrons act in ways that are different from everyday objects.

One important idea in quantum mechanics is how we picture changes over time. There are different ways to look at these changes, and each way helps us connect quantum ideas to the physics we see in the normal world. For example, there is a way called the phase space formulation that makes it easier to understand how quantum mechanics links to classic physics. Even though these ideas can get complex, they all help us learn about the hidden rules that control the tiny building blocks of our universe.

Problem of measurement

Main article: Measurement in quantum mechanics

Quantum mechanics is different from older science because it explains what happens when we look at or measure something. When we measure a quantum system, the result is random, but we can calculate the chances of different outcomes. For example, if we know the system is in a certain state, we can find out how likely it is that a measurement will show a specific value.

After a measurement, the system’s state changes. If we measure something many times in a row, we get the same result each time. This is an important idea in quantum mechanics. There are also newer ways to think about measurements, which help us describe many types of quantum actions using one framework.

List of mathematical tools

The book Methods of Mathematical Physics by Richard Courant, based on David Hilbert’s courses at Göttingen University, shows that math was ready for quantum mechanics before physicists needed it. When Erwin Schrödinger introduced his equation, they found the math they needed was already there.

Key math tools for quantum mechanics include: